John H. Frederick

Quantum mechanics in phase space: New approaches to the correspondence principle

We present a time evolution equation that provides a novel basis for the treatment of quantum systems in phase space and for the investigation of the quantum‐classical correspondence. Through the use of a generalized Husimi transform, we obtain a phase space representation of the time‐dependent Schrodinger equation directly from the coordinate representation. Such an equation governs the time evolution of densities such as the Husimi density entirely in phase space, without recourse to a coordinate or momentum representation. As an application of the phase‐space Schrodinger equation, we compute the eigenfunctions of the harmonic oscillator in phase space, relate these to the Husimi transform of coordinate representation eigenstates, and investigate the coherent state, its time evolution, and classical limit (ℏ→0) for the probability density generated by this state. Finally, we discuss our results as they relate to the quantum‐classical correspondence, and quasiclassical trajectory simulations of quantum d...

General formulation of the vibrational kinetic energy operator in internal bond-angle coordinates

A general formulation of the vibrational kinetic energy operator expressed in internal bond-angle coordinates is presented. This formulation is based on Podolsky’s expression for the covariant form of the Laplace–Beltrami operator. When a valid set of internal bond-angle coordinates is employed, it is possible to adapt a systematic approach to solve for the Jacobian determinant governing the coordinate transformation from Cartesian coordinates. In the general case of an arbitrary N-atom system, this Jacobian always factorizes to a simple form. This allows one to evaluate all the terms that contribute to V′, the effective potential that arises from transforming the kinetic energy operator to internal coordinates. We discuss restrictions on the choice of internal vibrational coordinates that may be included in a valid set. We then provide tabular information from which the vibrational kinetic energy operator for any molecular system can be constructed directly with no matrix inversion or chain rule manipul...

Classical trajectory study of vibration-rotation interaction in highly excited triatomic molecules

For two qualitatively different model triatomic molecules, SO2 and ‘‘bent equilibrium OCS,’’ classical trajectory calculations of the rotational and vibrational motion are presented for microcanonical ensembles of initial conditions at 35% to 85% of a bond dissociation energy. At the higher energies, many of the trajectories exhibit substantial intramolecular vibrational‐rotational energy transfer (IVRET), which has a significant effect on the geometry of rotational motion including in some cases, a transition between the two types of stable asymmetric top motion. IVRET is caused principally by centrifugal interactions, and in ‘‘bent OCS,’’ it is dominated by a 2:1 resonance. The rotational motion of about half of the high energy SO2 trajectories is essentially statistical, but bent OCS never undergoes statistical IVRET.

A quantum mechanical representation in phase space

A quantum mechanical representation suitable for studying the time evolution of quantum densities in phase space is proposed and examined in detail. This representation on L2 (2) phase space is based on definitions of the operators P and Q in phase space that satisfy various correspondences for the Liouville equation in classical and quantum phase space, as well as quantum position and momentum L2 (1) spaces. The definitions presented here, P=p/2−iℏ∂/∂q and Q=q/2+iℏ∂/∂p, are related to definitions that have been recently proposed [J. Chem. Phys. 93, 8862 (1990)]. The resulting quantum phase space representation shares many of the mathematical properties of usual representations in coordinate and momentum spaces. Within this representation, time evolution equations for complex‐valued functions (wave functions) and their square magnitudes (distribution functions) are derived, and it is shown that the coordinate and momentum space time evolution equations can be recovered by a simple Fourier projection. ...

Molecular Hamiltonians for highly constrained model systems

We describe a simple approach for constructing the molecular Hamiltonian which is ideal for studying large molecular systems in which many constraints are imposed. The present procedure allows one to evaluate the correct classical G matrix for highly constrained model systems in the absence of total angular momentum, using matrices that are no larger than the number of active vibrations in the model. A straightforward prescription for constructing the appropriate quantum kinetic energy operator for these systems is then introduced. This prescription is a modification of the Podolsky procedure and it allows one to incorporate constraints in the quantum operators without the extensive use of differential calculus. Finally, the extension of the constrained system Hamiltonian to nonzero total angular momentum is made and methods for reducing the effective Hamiltonian to the minimum number of degrees of freedom using the Augustin–Miller canonical transformation are described. The present approach is illustrate...

Multidimensional quantum dynamics with trajectories: a novel numerical implementation of Bohmian mechanics

Abstract A novel implementation of the de Broglie–Bohm mechanics is presented. The method employs the use of n-dimensional Delaunay tesselation for the purpose of computing the quantum potential term and is fully generalizable for the multidimensional case. We simulate the scattering of a Gaussian wavepacket from an Eckart barrier in two- and three-dimensions and compare our results against the dynamics obtained using a numerically exact propagation scheme.

Models for statistical decomposition of metal clusters: Vibrational frequency distributions

The application of statistical theories to the decomposition kinetics of metal clusters requires the estimation of the vibrational frequency distributions. We adapt elastic theories developed for bulk metals and fine particles to generate a physically reasonable frequency distribution model for small metal clusters. Results obtained from this elastic cluster model compare favorably with previously reported experimental heat capacity data for fine particles. In addition, predictions of the present model are shown to correlate very well with experimentally determined trends in metal cluster cohesive energies. The elastic cluster model is then applied to the statistical unimolecular decay kinetics of metal clusters and compared with results found using earlier theoretical models. The present model predicts slower rates of decomposition in comparison with the other models. These results suggest that the binding energies extracted from experimental photodissociation and collision‐induced dissociation measureme...

Stilbene isomerization dynamics on multidimensional potential energy surface. Molecular dynamics simulation

Abstract The dynamics in the photoisomerization of isolated cis-stilbene, especially the dynamics leading to the intermediate twisted configuration is explored by molecular dynamics simulation. Comparison of the dynamics of the trans and cis isomers shows that for cis-stilbene the time required to reach the twisted configuration is an order of magnitude or two less than for trans-stilbene, due only to the specific steric interactions, and not to any difference in the potential functions or parameters. A barrier of between 510 and 640 cm−1 along the torsional coordinate from the cis side is estimated for isolated molecule, accounting for a transition to the intermediate in approximately 300 fs.

S1–S2 vibronic coupling in trans-1,3,5-hexatriene. I. Electronic structure calculations

The estimates for the vertical excitation energy of the 2 1A1 state of cis-1,3,5-hexatriene (CHT) vary considerably and provide a good example of the difficulties that can arise in determining transition energies. The great uncertainty is surprising if one considers that this state has already been characterized by high resolution techniques such as resonance enhanced multiphoton ionization (REMPI) and fluorescence excitation spectroscopy in free jet expansions. A theoretical analysis of this problem is clearly needed and the present work, along with the following paper, represents an effort to investigate the nature of the 2 1A1 and 1 1B1 states of CHT. It is shown that a combination of ab initio electronic structure and quantum-mechanical wave packet calculations is required to systematically approach a question as involved as locating the energetical position of the 2 1A1 level. We characterize the energy dependence of the 1 1A1, 2 1A1, and 1 1B1 states of CHT as a function of the in-plane normal coord...

Nonlinear dynamics of torsion-rotation interactions: A model study of toluene

The internal rotation, or torsion, of a methyl group has been implicated in the acceleration of intramolecular vibrational redistribution (IVR) in numerous experimental studies. In the present work, we investigate its interaction with overall molecular rotation. To isolate the effects of torsion–rotation coupling, a simple two-degree-of-freedom model, including only torsion and three-dimensional rotation, is constructed and its dynamics at j=45 for several energies are studied. Investigation of other values of angular momentum indicate that the results reported are largely independent of j. Two primary effects are observed: (i) a shifting of the stable and unstable axes of rotation due to free methyl torsion, and (ii) a limited degree of weakly chaotic dynamics for trajectories whose torsional energy is near the top of its barrier. Chaos is first observed at the lowest energy at which torsion can surmount its barrier, but then disappears from the system at higher energies. Model toluene exhibits only narr...